Cluster size convergent full relativistic density-functional calculations of single atom adsorption

22 July 2002

Physics Letters A 300 (2002) 71–75 www.elsevier.com/locate/pla

Cluster size convergent full relativistic density-functional calculations of single atom adsorption T. Jacob a , S. Fritzsche a , W.-D. Sepp a , B. Fricke a,∗ , J. Anton b a Fachbereich Physik, Universität Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany b Department of Atomic Physics, Stockholm University, S-104 05 Stockholm, Sweden

Received 28 May 2002; received in revised form 6 June 2002; accepted 7 June 2002 Communicated by P.R. Holland

Abstract Within full relativistic four-component ab initio density functional cluster calculations we are now able to present results for the adsorption energy and bond distance of single atoms on surfaces. In order to compare with other (non-relativistic) calculations we have chosen Cu on a Cu(100) surface as an example. The convergence with respect to the cluster size is achieved with 56 atoms. This demonstrates that even for good conductors convergent molecular cluster calculations with reliable results are possible.  2002 Elsevier Science B.V. All rights reserved. PACS: 31.15.Ew; 31.15.Ne; 68.43.-h

For the understanding of adsorption processes the knowledge of the interaction between the adatom and the surface is indispensable. In contrast to very many solid-state properties the adsorption process is a local phenomenon. A number of ways exist to calculate these quantities with solid-state methods, but there is nearly no approach from the molecular side where convergence with cluster size is achieved. Within a full relativistic approach this will be investigated here for the first time. Ab initio density-functional calculations (DFT) provide a powerful tool to get a deeper insight into the microscopic regime not only for atoms and mole* Corresponding author.

E-mail addresses: [email protected] (T. Jacob), [email protected] (B. Fricke). URL address: http://www.physik.uni-kassel.de/theorie.

cules but also for solids [1]. It can also be used to describe problems in surface physics and catalysis on a quantum-chemical level [2]. To treat problems where only the surface is examined, one can utilize the translation symmetry by using the solid-state method of supercells or slabs [3,4]. The dominant problem in the supercell/slab-approach is the remaining interaction between the adsorbate atoms among each other and with the surface of the neighbor cells that are also repeated. If one increases the supercells to very large sizes, one will finally get converged values for the characteristic properties of the adsorption: adsorption energy and bond distance. Because every solid can be treated as a big molecule with very many atoms, methods usually used in cluster physics should also be valid for solids. But if one describes the surface by a limited cluster of atoms the remaining question is: how large should the clus-

0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 7 8 8 - 0

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T. Jacob et al. / Physics Letters A 300 (2002) 71–75

ter be chosen in order to describe the system properly? The cluster-size which is needed is strongly dependent on the elements that are involved. As an example, to describe the adsorption of CO on Pt(111) about 20 Pt atoms are sufficient [5], whereas Al on Al(100) requires much more atoms [6,7]. Especially those surfaces that are good conductors require a huge amount of atoms. The biggest calculations in the cluster approximation have been carried out for Au147 [8] or Pd309 [9] by Rösch et al. Their emphasis was to characterize the solid-state properties of these elements, where they were able to make use of the higher symmetry compared to the adsorption on a surface. They used a non-relativistic method with the addition of the scalar relativistic effects. In case of gold which is a 6selectron system, this probably is not too bad. For more complicated outer-shell systems in the region of heavy elements of course fully relativistic codes are indispensable. In this Letter we present cluster-size-dependent full relativistic calculations for the adsorption of Cu on Cu(100). We have chosen this non-relativistic system as a test because this allows a comparison with nonrelativistic calculations [10–13]. Copper itself is a well conducting material whose behavior is influenced by the 3d-band. The method we used is a full-relativistic four-component density functional approach. All the calculations were done self-consistently for all electrons in the system, which means that no further approximation was used. For a surface-cluster consisting of 60–70 atoms we found convergence in adsorption energy and bond distance. To verify the convergence we also calculated a system of 99 Cu atoms. For an estimation of the calculation expense, it should be remarked that it took about 9 months on a workstation to get one value of the potential energy curve of the largest (99 atoms) system. Only a heavy parallelization of the program finally made it possible to receive the results presented in this Letter. The method that is used for the calculations presented here has its roots in the relativistic form of the discrete variational method (DVM) [14]. Beginning with the no-pair approximation [15] and neglecting the minor important contributions from the spatial components of the four-current j ν (r ) = (j 0 , j ) the total energy of a system can be written as E[] = T S + E N [] + E C [] + E xc []

(1)

with the electronic density 



 ψi+ r ψi r . j 0 r =  r = −mc2 <ε

(2)

i
εF

Here the density is obtained from a sum of M auxiliary one-particle Dirac spinors. Then the corresponding relativistic form of the Kohn–Sham equations (rKS) [16,17] is





tˆ + V N r + V C r + V xc r ψi r = εi ψi r , (3)  + (β − 1)mc2 is the Dirac kinetic where tˆ = −ic α·∇ energy operator and V N and V C are the Coulomb potentials of the nuclei and the electrons. εi are the energy eigenvalues of the Dirac spinors and V xc is the exchange–correlation potential defined by
 δE xc [(r )] V xc r = . δ(r )

(4)

During the self-consistent cycles the relativistic localdensity approximation (rLDA) [16,18] for the exchange–correlation functional is applied together with a parametrization for the correlation suggested by Vosko, Wilk, and Nusair [19]. Finally the total energy is calculated perturbatively with the relativistic forms of the generalized gradient approximation (rGGA) [20]. For the exchange we use Becke’s [21] formulation and for the correlation the functional proposed by Perdew [22] (B88/P86) Details of the computational method can be found in Refs. [23–26]. As an additional aspect it should be mentioned that the code was parallelized using MPI (message passing interface) [27]. To examine the behavior of the convergence in binding energy and distance we started with a cluster of 5 atoms where four copper atoms represent the Cu(100) surface and then increased the size of the cluster up to 99 atoms. All systems were treated full self-consistently in all electron calculations: Cu–Cu4 ,

Cu–Cu4.5 ,

Cu–Cu16.13.12.1, Cu–Cu24.21.16.9

Cu–Cu12.9.4 ,

Cu–Cu16.21.12.5, and Cu–Cu32.25.24.13.4

(see Fig. 1). In this notation the indices give the number of atoms per layer. Thus the largest system spans over 5 layers in vertical direction. The adatom was always located at the four-fold position on the

T. Jacob et al. / Physics Letters A 300 (2002) 71–75

Fig. 1. Model of the system Cu–Cu32.25.24.13.4 . The adsorbate (dark) is placed at the four-fold site on the surface (grey). In lateral direction the third nearest neighbors are included, and in vertical direction the cluster spans over 5 layers.

Cu(100) surface-cluster, and the Cu–Cu bond distance of the surface was set to the bulk value of 2.553 Å. The atomic basis set used for all Cu atoms contains 1s, 2s, 2p1/2, 2p3/2, 3s, 3p1/2, 3p3/2, 3d3/2, 3d5/2, 4s, 4p1/2 and 4p3/2 wave functions. Of course, we know that a “better” basis should include more polarization functions. But a further extension of the basis increases the computational effort and makes the

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practical calculation impossible. As mentioned above, the calculation of the potential energy surface of the largest system with nearly 100 atoms took about 9 months, for example. Our experience with a similar system (Al on Al(100)) [7,28] showed that the size of the basis set becomes less important with increasing size of the clusters. Therefore we assume that this also holds true for the present system. The potential energy curves of all calculated systems are shown in Fig. 2. For clarity only those adsorption energies are shown that were calculated with the rLDA-functional for the exchange correlation. In Table 1 we also list the adsorption energies which we got with the gradient corrected GGA-functionals B88/P86 [21,22]. Comparison is given with the nonrelativistic slab-calculations of Pentcheva [10] and Shin and Scheffler [11], and with the embedded-atom results of Kürpick [12,13]. In their slab-calculations Pentcheva and Shin used an unit cell of about 60 atoms that was repeated periodically [29]. The two different results of Kürpick base on different embedding potentials for the element copper. Comparing the runs of all potential energy curves in Fig. 2 one can perceive that the behavior of each

Fig. 2. Potential energy curves for all calculated systems. Only the energies for the exchange–correlation functional in rLDA-approximation are shown. The final values have to be corrected for the GGA contribution (see Table 1).

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T. Jacob et al. / Physics Letters A 300 (2002) 71–75

Table 1 Binding energies and distances (only rLDA) for all calculated systems. The adsorption energies that are calculated within the rLDA- and B88/P86-functional are printed. Additionally the vibration frequencies of the adsorbate are given that were calculated by fitted Morse potentials (the error is maximum 5%) System

Adsorption energy (eV)

Bond distance (a.u.)

Vibration frequency (cm−1 )

rLDA

B88/P86

rLDA

rLDA

Cu–Cu4 Cu–Cu4.5 Cu–Cu12.9.4 Cu–Cu16.13.12.1 Cu–Cu16.21.12.5 Cu–Cu24.21.16.9 Cu–Cu32.25.24.13.4

−3.60 −3.52 −2.75 −3.18 −2.59 −2.70 −2.70

−2.55 −2.64 −1.84 −2.46 −1.68 −1.79 −1.79

2.70 2.95 2.95 3.20 3.25 3.35 3.34

165 166 153 154 153 154 154

Pentcheva [10] Shin [11] Kürpick [12] Kürpick [13]

−2.84 −2.74 −2.91 −2.80

curve near the minimum is nearly the same. Only the positions of the minima are displaced. That means that even small clusters are sufficient for reliable results of the vibration frequency (see Table 1). If the size of the surface-cluster is increased up to 26 atoms successively the adsorption energy decreases. Further expansion to 43 atoms forces a stronger binding. With 55 atoms the adsorption energy decreases again, but the converged values are nearly reached. The last two expansions to 71 and 99 atoms show only very little change in the energy value. Hence −2.70 eV is the converged rLDA-value of the adsorption energy. Comparison with the results of the other groups [10–13] that used the LDAapproximation shows a good agreement. The best correspondence was obtained with the value calculated by Shin. Among the calculations of Pentcheva and Shin who used nearly the same method Shin’s result is more recent. The difference between Kürpick’s values and our result is 0.1–0.2 eV. But compared to the error bar of the embedding-atom values caused by the choice of the Cu embedding-potential, the agreement seems to be quite well. During the convergence in size it is observed that a lateral extension increases and a vertical extension decreases the adsorption energy. Of course, after reaching convergence this effect is not recognizable any more. In bond distance there seems to be no correlation with the direction of cluster-expansion. There is nearly

3.15 3.08 2.97

a continuous increase in the values for the bond distance when enlarging the surface-cluster. Equivalent to the adsorption energy, the convergence in bond distance is reached with about 55 atoms, but with an error bar of about 0.1 a.u. For the extended systems (71 and 99 atoms) the error is reduced to ≈ 0.01 a.u. Finally the converged bond distance is 3.34 a.u. Comparison to the results of other groups shows a difference of 0.2–0.3 a.u., which is easily explained by the relaxation effects that their calculations can consider more easily than our method. The nearest neighbors of the adatom are somewhat pushed into the bulk. Therefore the bond distance of the adatom is somewhat smaller than the value we calculated. Within the GGA-functional the bond distances for the smaller clusters were about 0.3 a.u. larger compared to the rLDA-values. This behavior is caused by the weaker binding which one gets from the GGA-calculations. But if the cluster is enlarged this difference decreases. For the 71 and 99 atomic systems the difference in bond lengths for both functionals is
0.01 a.u. Up to now we have not found an easy explanation for this. In summary, one can point out that the adsorption process of Cu on the Cu(100) surface can already be simulated by a cluster of 60–70 atoms very well. Further increase showed no effect, and thus convergence was achieved. In the rLDA approximation of the exchange–correlation term the converged binding energy is −2.70 eV and the bond distance is 3.34 a.u. The inclusion of the GGA-functional changes

T. Jacob et al. / Physics Letters A 300 (2002) 71–75

the binding energy to −1.79 eV which should be compared to the experiment. For the large clusters very little changes in bond distance compared to rLDA. were observed. The calculations that were presented show that molecular methods are also able to give reliable results for adsorption processes. Therefore, we will use our full relativistic method to treat similar problems for very heavy systems in the future.

Acknowledgements T.J. gratefully acknowledges support by the Studienstiftung des Deutschen Volkes, J.A. by the Gesellschaft für Schwerionenforschung (GSI) and DFG.

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